How can one justify Newton's third law?

Question

In one sense it is justified by the overall success if Newtonian Mechanics; still, one can ask are there arguments that can justify it from other principles; ie principle that are * a priori* in nature. For example, Kant supplied one, in his Metaphysics of Natural Science; and to which he added the remark:

“[Newton] by no means dared to prove this law a priori, and therefore appealed rather to experience”

His argument is as follows:

(i) if all changes of matter are changes of motion;

(ii) if all changes of motion are reciprocal and equal (since one body cannot move closer to/farther away from another body without the second body moving closer to/farther away from the first body and by exactly the same amount); and

(iii) if every change of matter has an external cause (a proposition that was established as the Second Law of Mechanics), then the cause of the change of motion of the one body entails an equal and opposite cause of a change of motion of the other body or, in short, action must be equal to reaction

Is the following argument more basic? In that (ii) is deduced from considerations of symmetry:

Consider an action: what does this mean? A substance can't act on itself, for how can we say it acts now as opposed to then? To make this concrete, consider a classical electron - its negative charge doesn't act on itself - otherwise it wouldn't have any cohesion.

It acts then, on an other; and then on contact; but by symmetry, ie swapping the one substance with the other, the same situation is obtained.

Hence every action has an equal an opposite reaction

Where equal and opposite are not to be understood in quantifiable terms; but in the terms outlined above.

And this argued as an outcome of action by contact; and by contact that one is simultaneous in place and time with another; and that this relation is symmetric.

Answer 1

See Noether's Theorem.

This theorem states that for every symmetry, there must be a corresponding conserved quantity. It's math.

The universe is symmetric regarding translations in space (if everything were moved three feet to the left, nothing would change). The conserved quantity for this symmetry is momentum. Thus, because the universe is symmetric regarding translations, momentum must be conserved.

"For every action, there is a (quantifiably) equal and opposite reaction" is logically equivalent to "momentum is a conserved quantity."

Newton justified his law as a generalization of many observations. We justify it today because Noether's Theorem says it must be true.

Answer 2

A. P. French wrote in 1971 in his book "Newtonian Mechanics", page 317 and figure 9-4, that if you consider a gun firing particles analogous to photons at a barrier some distance away, that since the photon-like objects cannot be seen, there is the appearance that Newton's third law does not apply until it is absorbed by the barrier.

Scientifically we say that photons have momentum, and we could measure it anywhere along the path from the gun to the barrier, however the issue is that a photon does not take a quantifiable state until such an absorption occurs. That is to say, it defers to the philosophical question of "Where are photons between emission and absorption?" (Which does not have a precise philosophical answer in the context of the double-slit experiment).

Answer 3

Kant's argument is incorrect. The following statement is false:

if all changes of motion are reciprocal and equal (since one body cannot move closer to/farther away from another body without the second body moving closer to/farther away from the first body and by exactly the same amount);

If a body of $1$ $\mathrm{kg}$ interacts (e.g. via a spring, or electromagnetic force) with a body of $2$ $\mathrm{kg}, then the change in velocity of the first body will be double the change of velocity in the second body. Any correct derivation of Newton's third law must certainly involve mass, or otherwise it must only apply to bodies of equal mass.

The argument remains incorrect if we consider two equal bodies, because it confuses change in position with change in motion, and it confuses relative motion with motion with respect to a reference frame. Consider a system of many objects of equal mass in any motion whatsoever, and let $A$ and $B$ be two of the objects The assumption that if $A$ moves some amount relative to $B$, then $B$ must move the opposite of that amount relative to $A$, remains true. However, an arbitrary motion of a system of objects certainly does not satisfy Newton's third law. (If it did, then the law would also be vacuous.)

In order to form a correct argument one must first clearly define terms such as force, mass, position, velocity, acceleration. Newton's third law says that the forces are equal and opposite. Neither Kant's argument nor your argument mentions the word force.

Remember that Newton's third law is not just a law about two identical objects. It is a law about two different objects interacting in any manner whatsoever. So the symmetry argument does not work either.

In physics we define force as the time derivative of momentum. So if $p_A$ is the momentum of object $A$ and $p_B$ is the momentum of object $B$, then Newton's third law says $$\frac{dp_A}{dt}=- \frac{dp_B}{dt}$$ We can rewrite this as $$\frac{dp_A}{dt} + \frac{dp_B}{dt} = 0$$ Or, defining the total momentum $P=p_A+p_B$, the law states $dP/dt=0$. This is conservation of momentum. Conservation of momentum is a consequence of two facts: (1) particles move in trajectories of least action (2) the action is invariant under space translation. (1) is a consequence of quantum mechanics, and (2) is an experimental fact. That conservation of momentum follows from (1) and (2) is called Noether's theorem. You say that "I'm not interested in a justification by recourse to Noether's theorem", but Noether's theorem is the justification. You can surely come up with a string of words that tickles some human brains in just the right way as to activate the "I'm convinced by this argument" center, but that string of words will not work on physicists. You can't do physics without doing physics.

Answer 4

1) Can you explain to me how to apply Kant's principles to the following scenario? Suppose two objects A and B, and that B is pushed towards A by some external cause; suppose we neglect gravity and electrical attraction between A and B; according to Newton there is no action and reaction between A and B (until they collide), is there? but according to Kant, there is, isn't there? what is going on?

Is there a consensus that Kant's and Newton's laws are equivalent?

2) As an answer to your question, Feynman derives conservation of momentum (in collisions) from Galilean relativity in Chapter 10 of the Feynman Lectures, and a few simple a posteriori assumptions.

On the other hand he argues that even Galilean relativity (and physics in general) is not a priori in chapter 16:

Our inability to detect absolute motion is a result of experiment and not a result of plain thought

He then presents an interesting argument to support that statement.

So, is there a consensus among philosophers of science and physicists that Kant's laws of motion are a priori?